# Sharkovskii’s theorem

Every natural number  can be written as $2^{r}p$, where $p$ is odd, and $r$ is the maximum exponent  such that $2^{r}$ divides (http://planetmath.org/Divisibility) the given number. We define the Sharkovskii ordering of the natural numbers in this way: given two odd numbers   $p$ and $q$, and two nonnegative integers $r$ and $s$, then $2^{r}p\succ 2^{s}q$ if

1. 1.

$r and $p>1$;

2. 2.

$r=s$ and $p;

3. 3.

$r>s$ and $p=q=1$.

This defines a linear ordering of $\mathbb{N}$, in which we first have $3,5,7,\dots$, followed by $2\cdot 3$, $2\cdot 5,\dots$, followed by $2^{2}\cdot 3$, $2^{2}\cdot 5,\dots$, and so on, and finally $2^{n+1},2^{n},\dots,2,1$. So it looks like this:

 $3\succ 5\succ\cdots\succ 3\cdot 2\succ 5\cdot 2\succ\cdots\succ 3\cdot 2^{n}% \succ 5\cdot 2^{n}\succ\cdots\succ 2^{2}\succ 2\succ 1.$

Sharkovskii’s theorem. Let $I\subset\mathbb{R}$ be an interval, and let $f:I\rightarrow\mathbb{R}$ be a continuous function   . If $f$ has a periodic point  of least period $n$, then $f$ has a periodic point of least period $k$, for each $k$ such that $n\succ k$.

Title Sharkovskii’s theorem SharkovskiisTheorem 2013-03-22 13:16:11 2013-03-22 13:16:11 Koro (127) Koro (127) 7 Koro (127) Definition msc 37E05 Sharkovsky’s theorem Sharkovskii’s ordering Sharkovsky’s theorem